Comparison and union of the Temple and Bazley lower bounds

Marmorino, M. G.

doi:10.1007/s10910-013-0199-7

Cited by 7 publications

(7 citation statements)

References 5 publications

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“…The H jj matrix elements are then arranged in ascending order, and one may construct with them a matrix Hamiltonian just as in eq 2. 29, by replacing the Ritz eigenvalues and associated variances with the H jj and associated σ jj values. The PM equation, eq 2.30, remains the same, the main difference is that, apart from the ground-state matrix element H 11 (and all other diagonal matrix elements) which is always greater than or equal to the true ground-state energy, it is no longer necessarily the case that all other diagonal elements bound the excited state eigenvalues from above.…”

Section: Pm Lower Bound Theorymentioning

confidence: 99%

“…Their energies bound the exact eigenvalues from below. Further analysis and improvements upon Bazley’s method as well as comparisons with Temple-based lower bounds have been presented by Marmorino. − However, here too the bottom line is not very encouraging. The complete basis set of the “base” separable Hamiltonian is not complete for a multielectron atom so that the method will not necessarily converge to the exact answer.…”

Section: Introductionmentioning

confidence: 99%

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Lower Bounds for Nonrelativistic Atomic Energies

et al. 2021

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A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021Comput. , 17, 1535 is further developed and applied to the highly accurate calculation of the ground-state energy of two-(He, Li + , and H − ) and three-(Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained.Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.

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Section: Pm Lower Bound Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lower Bounds for Nonrelativistic Atomic Energies

et al. 2021

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“…Lower bound methods abound, starting with Temple’s seminal expression derived in 1928 . Landmarks in the derivation of lower bounds are Weinstein’s lower bound of 1934 , and Lehmann’s optimization of Temple’s lower bound presented in 1949–50. , Especially Lehmann’s expression has turned out to be quite accurate in different settings, however not so for Coulombic systems, as exemplified by computations on the He − and Li atoms.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds for Coulombic Systems

Pollak

Martinazzo

2021

J. Chem. Theory Comput.

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As of the writing of this paper, lower bounds are not a staple of quantum chemistry computations and for good reason. All previous attempts at applying lower bound theory to Coulombic systems led to lower bounds whose quality was inferior to the Ritz upper bounds so that their added value was minimal. Even our recent improvements upon Temple’s lower bound theory were limited to Lanczos basis sets and these are not available to atoms and molecules due to the Coulomb singularity. In the present paper, we overcome these problems by deriving a rather simple eigenvalue equation whose roots, under appropriate conditions, give lower bounds which are competitive with the Ritz upper bounds. The input for the theory is the Ritz eigenvalues and their variances; there is no need to compute the full matrix of the squared Hamiltonian. Along the way, we present a Cauchy–Schwartz inequality which underlies many aspects of lower bound theory. We also show that within the matrix Hamiltonian theory used here, the methods of Lehmann and our recent self-consistent lower bound theory ( J. Chem. Phys. 2020, 115, 244110) are identical. Examples include implementation to the hydrogen and helium atoms.

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“…24,25 These approaches were first applied to He energy levels using a special choice by Bazley and Fox, 26,27 and motivated several further improvements for Coulombic and other potentials. [28][29][30][31][32][33] Bracketing functions have also been successfully applied to lower bound problems. [34][35][36][37][38] Further lower bound calculation strategies are also available for He atoms.…”

Section: Introductionmentioning

confidence: 99%

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

Rontó

Pollak

Martinazzo

2021

Preprint

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Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the SCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.

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Comparison and union of the Temple and Bazley lower bounds

Cited by 7 publications

References 5 publications

Lower Bounds for Nonrelativistic Atomic Energies

Lower Bounds for Nonrelativistic Atomic Energies

Lower Bounds for Coulombic Systems

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

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